The Euclidean two-point correlation function of the topological charge density

We study the Euclidean two-point correlation function $G_q(x)$ of the topological charge density in QCD. A general statement based on reflection positivity tells us that $G_q(x)<0$ for $x\mathrm{\ne }0$. On the other hand the topological susceptibility ${\chi }_q=\int d^{d}x{G}_q(x)$ is a positive quantity. This indicates that $G_q(x)$ developes a positive contact term at $x=0$, that contributes to the determination of the physical value of ${\chi }_q$. We show explicitly these features of $G_q(x)$ in a solvable nontrivial continuum model, the two-dimensional $C{P}^{N-1}$ model in the large-N limit. A similar analysis is done on the lattice.